| L(s) = 1 | + 208·2-s − 4.37e3·3-s − 1.28e4·4-s + 1.56e5·5-s − 9.09e5·6-s + 1.95e5·7-s − 7.53e6·8-s + 1.43e7·9-s + 3.25e7·10-s − 1.26e7·11-s + 5.62e7·12-s − 2.72e8·13-s + 4.07e7·14-s − 6.83e8·15-s − 5.01e8·16-s − 2.70e9·17-s + 2.98e9·18-s − 2.24e9·19-s − 2.01e9·20-s − 8.56e8·21-s − 2.63e9·22-s − 1.62e10·23-s + 3.29e10·24-s + 1.83e10·25-s − 5.66e10·26-s − 4.18e10·27-s − 2.51e9·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.15·3-s − 0.392·4-s + 0.894·5-s − 1.32·6-s + 0.0898·7-s − 1.27·8-s + 9-s + 1.02·10-s − 0.196·11-s + 0.453·12-s − 1.20·13-s + 0.103·14-s − 1.03·15-s − 0.466·16-s − 1.60·17-s + 1.14·18-s − 0.576·19-s − 0.351·20-s − 0.103·21-s − 0.225·22-s − 0.992·23-s + 1.46·24-s + 3/5·25-s − 1.38·26-s − 0.769·27-s − 0.0352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+15/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p^{7} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{7} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 13 p^{4} T + 877 p^{6} T^{2} - 13 p^{19} T^{3} + p^{30} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 27968 p T + 10537896002 p^{3} T^{2} - 27968 p^{16} T^{3} + p^{30} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 12677912 T + 8233055677242454 T^{2} + 12677912 p^{15} T^{3} + p^{30} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 272571732 T + 9120511586549542 p T^{2} + 272571732 p^{15} T^{3} + p^{30} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2709086588 T + 7398134051960456422 T^{2} + 2709086588 p^{15} T^{3} + p^{30} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2248025064 T + 25566844129032196678 T^{2} + 2248025064 p^{15} T^{3} + p^{30} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16214470848 T + \)\(28\!\cdots\!14\)\( T^{2} + 16214470848 p^{15} T^{3} + p^{30} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 167604104788 T + \)\(20\!\cdots\!58\)\( T^{2} + 167604104788 p^{15} T^{3} + p^{30} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 244622120128 T + \)\(39\!\cdots\!02\)\( T^{2} + 244622120128 p^{15} T^{3} + p^{30} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 1055700361276 T + \)\(76\!\cdots\!06\)\( T^{2} - 1055700361276 p^{15} T^{3} + p^{30} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 81131618188 T + \)\(30\!\cdots\!22\)\( T^{2} + 81131618188 p^{15} T^{3} + p^{30} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1312122690840 T + \)\(37\!\cdots\!50\)\( T^{2} - 1312122690840 p^{15} T^{3} + p^{30} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7338061052080 T + \)\(37\!\cdots\!90\)\( T^{2} - 7338061052080 p^{15} T^{3} + p^{30} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3755998579268 T - \)\(48\!\cdots\!26\)\( T^{2} + 3755998579268 p^{15} T^{3} + p^{30} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11942083041176 T + \)\(31\!\cdots\!98\)\( T^{2} + 11942083041176 p^{15} T^{3} + p^{30} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18080697194580 T + \)\(47\!\cdots\!98\)\( T^{2} + 18080697194580 p^{15} T^{3} + p^{30} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 50227778651128 T + \)\(31\!\cdots\!82\)\( T^{2} + 50227778651128 p^{15} T^{3} + p^{30} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 66695913482384 T + \)\(11\!\cdots\!66\)\( T^{2} - 66695913482384 p^{15} T^{3} + p^{30} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 232554112724748 T + \)\(29\!\cdots\!54\)\( T^{2} + 232554112724748 p^{15} T^{3} + p^{30} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 161119142079040 T + \)\(24\!\cdots\!98\)\( T^{2} + 161119142079040 p^{15} T^{3} + p^{30} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 154296276009096 T + \)\(12\!\cdots\!42\)\( T^{2} - 154296276009096 p^{15} T^{3} + p^{30} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 305587106859156 T + \)\(30\!\cdots\!78\)\( T^{2} - 305587106859156 p^{15} T^{3} + p^{30} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1018905479409788 T + \)\(10\!\cdots\!22\)\( T^{2} + 1018905479409788 p^{15} T^{3} + p^{30} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.81953629468194484886232846007, −14.76841458057856019279146049653, −13.65837400195428653460441011342, −13.24060517185236952473918890471, −12.70041022940421114285601803187, −12.14349001510998590782055663383, −11.09175171929152497456247288668, −10.60699125417960805831934055930, −9.404664273946861276236804334265, −9.231777574831955358237661083846, −7.68126753524905242850848310337, −6.73054131979109285138442670061, −5.67591080802224730428673600027, −5.55148641879704667252603668881, −4.37370433069267140213587321096, −4.22251643733375301707646102639, −2.55757358276489139157861516745, −1.68226201882518838665386767431, 0, 0,
1.68226201882518838665386767431, 2.55757358276489139157861516745, 4.22251643733375301707646102639, 4.37370433069267140213587321096, 5.55148641879704667252603668881, 5.67591080802224730428673600027, 6.73054131979109285138442670061, 7.68126753524905242850848310337, 9.231777574831955358237661083846, 9.404664273946861276236804334265, 10.60699125417960805831934055930, 11.09175171929152497456247288668, 12.14349001510998590782055663383, 12.70041022940421114285601803187, 13.24060517185236952473918890471, 13.65837400195428653460441011342, 14.76841458057856019279146049653, 14.81953629468194484886232846007
